Results 1  10
of
23
A Conjectured Combinatorial Formula for the Hilbert . . .
, 2004
"... We introduce a conjectured way of expressing the Hilbert series of diagonal harmonics as a weighted sum over parking functions. Our conjecture is based on a pair of statistics for the q,tCatalan sequence discovered by M. Haiman and proven by the first author and A. Garsia (Proc. Nat. Acad. Sci. 98 ..."
Abstract

Cited by 24 (12 self)
 Add to MetaCart
We introduce a conjectured way of expressing the Hilbert series of diagonal harmonics as a weighted sum over parking functions. Our conjecture is based on a pair of statistics for the q,tCatalan sequence discovered by M. Haiman and proven by the first author and A. Garsia (Proc. Nat. Acad. Sci. 98 (2001), 43134316). We show how our q,tparking function formula for the Hilbert series can be expressed more compactly as a sum over permutations. We also derive two equivalent forms of our conjecture, one of which is based on the original pair of statistics for the q,tCatalan introduced by the first author and the other of which is expressed in terms of rooted, labelled trees.
An Introduction to Hyperplane Arrangements
 Lecture notes, IAS/Park City Mathematics Institute
, 2004
"... ..."
Parking Functions, Empirical Processes, and the Width of Rooted Labeled Trees
"... This paper provides tight bounds for the moments of the width of rooted labeled trees with n nodes, answering an open question of Odlyzko and Wilf (1987). To this aim, we use one of the many onetoone correspondences between trees and parking functions, and also a precise coupling between parking f ..."
Abstract

Cited by 20 (5 self)
 Add to MetaCart
This paper provides tight bounds for the moments of the width of rooted labeled trees with n nodes, answering an open question of Odlyzko and Wilf (1987). To this aim, we use one of the many onetoone correspondences between trees and parking functions, and also a precise coupling between parking functions and the empirical processes of mathematical statistics. Our result turns out to be a consequence of the strong convergence of empirical processes to the Brownian bridge (Komlos, Major and Tusnady, 1975).
k, m)Catalan numbers and hook length polynomials for trees
"... Abstract: In this paper we define and study the (k,m)Catalan number Ck,m(n) = 1 mn+1 n which is a generalization of the Catalan number C(n) = 1 2n n+1 n, and give two combinatorial interpretations for this number: (k,m)ary trees with n crucial vertices and (mn+1)×k “parking tables”. Using these ..."
Abstract

Cited by 14 (1 self)
 Add to MetaCart
Abstract: In this paper we define and study the (k,m)Catalan number Ck,m(n) = 1 mn+1 n which is a generalization of the Catalan number C(n) = 1 2n n+1 n, and give two combinatorial interpretations for this number: (k,m)ary trees with n crucial vertices and (mn+1)×k “parking tables”. Using these interpretations we give recurrence formulas for Ck,m(n) which can be
On the Sandpile Group of a Graph
 European Journal of Combinatorics
, 2000
"... We show how to express the sandpile model, introduced in theoretical physics, using the vocabulary of combinatorial theory. The group of recurrent configurations in the sandpile model, introduced by D. Dhar ([6]), may be considered as a finite abelian group associated with any graph G; we call it th ..."
Abstract

Cited by 12 (1 self)
 Add to MetaCart
We show how to express the sandpile model, introduced in theoretical physics, using the vocabulary of combinatorial theory. The group of recurrent configurations in the sandpile model, introduced by D. Dhar ([6]), may be considered as a finite abelian group associated with any graph G; we call it the sandpile group of G. The structure of the sandpile group is determined for some families of graphs.
A simple bijection for the regions of the Shi arrangement of hyperplanes
, 1997
"... The Shi arrangement Sn is the arrangement of affine hyperplanes in R n of the form xi−xj = 0 or 1, for 1 ≤ i < j ≤ n. It dissects R n into (n+1) n−1 regions, as was first proved by Shi. We give a simple bijective proof of this result. Our bijection generalizes easily to any subarrangement of Sn cont ..."
Abstract

Cited by 10 (2 self)
 Add to MetaCart
The Shi arrangement Sn is the arrangement of affine hyperplanes in R n of the form xi−xj = 0 or 1, for 1 ≤ i < j ≤ n. It dissects R n into (n+1) n−1 regions, as was first proved by Shi. We give a simple bijective proof of this result. Our bijection generalizes easily to any subarrangement of Sn containing the hyperplanes xi − xj = 0 and to the extended Shi arrangements.
Conjectured combinatorial models for the Hilbert series of generalized diagonal harmonics modules
 Electron.J.Combin.11 (2004), R68; 64
"... Haglund and Loehr previously conjectured two equivalent combinatorial formulas for the Hilbert series of the GarsiaHaiman diagonal harmonics modules. These formulas involve weighted sums of labelled Dyck paths (or parking functions) relative to suitable statistics. This article introduces a third c ..."
Abstract

Cited by 6 (4 self)
 Add to MetaCart
Haglund and Loehr previously conjectured two equivalent combinatorial formulas for the Hilbert series of the GarsiaHaiman diagonal harmonics modules. These formulas involve weighted sums of labelled Dyck paths (or parking functions) relative to suitable statistics. This article introduces a third combinatorial formula that is shown to be equivalent to the first two. We show that the four statistics on labelled Dyck paths appearing in these formulas all have the same univariate distribution, which settles an earlier question of Haglund and Loehr. We then introduce analogous statistics on other collections of labelled lattice paths contained in trapezoids. We obtain a fermionic formula for the generating function for these statistics. We give bijective proofs of the equivalence of several forms of this generating function. These bijections imply that all the new statistics have the same univariate distribution. Using these new statistics, we conjecture combinatorial formulas for the Hilbert series of certain generalizations of the diagonal harmonics modules. 1
Counting Defective Parking Functions
, 803
"... Suppose that m drivers each choose a preferred parking space in a linear car park with n spaces. Each driver goes to the chosen space and parks there if it is free, and otherwise takes the first available space with a larger number (if any). If all drivers park successfully, the sequence of choices ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
Suppose that m drivers each choose a preferred parking space in a linear car park with n spaces. Each driver goes to the chosen space and parks there if it is free, and otherwise takes the first available space with a larger number (if any). If all drivers park successfully, the sequence of choices is called a parking function. In general, if k drivers fail to park, we have a defective parking function of defect k. Let cp(n,m,k) be the number of such functions. In this paper, we establish a recurrence relation for the numbers cp(n,m,k), and express this as an equation for a threevariable generating function. We solve this equation using the kernel method, and extract the coefficients explicitly: it turns out that the cumulative totals are partial sums in Abel’s binomial identity. Finally, we compute the asymptotics of cp(n,m,k). In particular, for the case m = n, if choices are made independently at random, the limiting distribution of the defect (the number of drivers who fail to park), scaled by the square root of n, is the Rayleigh distribution. On the other hand, in the case m = ω(n), the probability that all spaces are occupied tends asymptotically to one. 1
Enumeration Of Trees And One Amazing Representation Of The Symmetric Group
 University of Minnesota
, 1995
"... . In this paper we present a systematic approach to enumeration of different classes of trees and their generalizations. The principal idea is finding a bijection between these trees and some classes of Young diagrams or Young tableaux. The latter arise from the remarkable representation of the symm ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
. In this paper we present a systematic approach to enumeration of different classes of trees and their generalizations. The principal idea is finding a bijection between these trees and some classes of Young diagrams or Young tableaux. The latter arise from the remarkable representation of the symmetric group studied by Haiman in connection with diagonal harmonics (see [7]). Define the vector space V ' hx a 1 oe(1) : : : x an oe(n) joe 2 Sn ; 0 a i i \Gamma 1; 1 i ni. Let the symmetric group Sn act on Vn by the permutation of variables. It is known that dim(Vn ) = (n + 1) n\Gamma1 is equal to the number of labeled trees, and dim(Vn ) Sn = 1 n+1 \Gamma 2n n \Delta is equal to the number of plane trees with n vertices. There are combinatorial interpretations for the other multiplicities. We generalize all the results in case of kdimensional trees and (k + 1)ary trees. 1. Introduction. Define a vector space (11) V ' hx a1 oe(1) : : : x an oe(n) j oe 2 Sn ; 0 ...